Question

In the given figure, if P, Q, R and S are
the mid-points of sides AB, AD, CD,
BC respectively, prove that PORS is a
parallelogram.
​

Answer 1

Answer:

Given, P, Q, R and S are the mid-points of sides AB, AD, CD, and BC respectively.

Now, join AC, BD, PS, QR, PQ and RS

Since, P is the mid point of AB

So, AP = PB  

Since, Q is the mid point of BC

So, QC = QB  

Since, R is the mid point of CD

So, CR = RD  

Since, S is the mid point of AD

So, AS = SD  

Now, divide equation 1 by equation 4, we get

AP/AS = PB/SD

=> AP/PB = AS/SD

=> PS || BD  {converse of Thales Theorem}

Similarly, QR || BD  

Again, From equation 5 and 6, we get

PS || QR  

Now, divide equation 1 by equation 2, we get

AP/QC = PB/QB

=> AP/PB = QC/QB

=> PQ || AC  {converse of Thales Theorem}

Similarly, SR || AC  

From 8 and 9, PQ || SR  

From 7 and 10, PS | QR and PQ || SR

Hence, PQRS is a parallelogram

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